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MathQuiz 4.3
University of Sydney
School of Mathematics
and Statistics
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Anti-differentiation (analytically) Quiz

Question

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This quiz tests the work covered in the lecture on the analytical interpretation of the anti-derivative and corresponds to Section 6.2 of the textbook Calculus: Single and Multivariable (Hughes-Hallett, Gleason, McCallum et al.).
There are more web quizzes at Wiley, select Section 2.

There are quizzes and web links on this topic at http://quiz.econ.usyd.edu.au/mathquiz/integration/index.php

Question 1

Which of the following statements are correct?

There is at least one mistake.
For example, choice (a) should be false.

$2x$ is the derivative of $x2 .$

There is at least one mistake.
For example, choice (b) should be true.

$( ) -d- 1x4 + c = 4x3 = 2x3 dx 2 2$

There is at least one mistake.
For example, choice (c) should be false.

$( ) -d- 1 3 2 2 dx 3x + 3x + x+ c = x +6x + 1$ so the statement is incorrect.

There is at least one mistake.
For example, choice (d) should be false.

$d-- dx (cosx +c) = - sinx$ so the statement is incorrect.

There is at least one mistake.
For example, choice (e) should be true.

$d-(sin x+ c) = cosx dx$

Your answers are correct

False. $2x$ is the derivative of $x2 .$
True. $( ) -d- 1x4 + c = 4x3 = 2x3 dx 2 2$
False. $( ) -d- 1 3 2 2 dx 3x + 3x + x+ c = x +6x + 1$ so the statement is incorrect.
False. $d-- dx (cosx +c) = - sinx$ so the statement is incorrect.
True. $d-(sin x+ c) = cosx dx$

Question 2

Which of the following correctly gives the anti-derivative $F (x)$ with $F ′(x) = 2 sinx + 2x2 + 5x+ 1$ where $F (0) = 2$ ?

Not correct. Choice (a) is false.

Try again, the derivative of this function is $F′(x) = - 2 sinx + 3x2 + 5x+ 1 .$

Not correct. Choice (b) is false.

Try again, the derivative of this function is $F′(x) = 2sin x+ x2 +x + 1.$

Not correct. Choice (c) is false.

Try again, $F(0) = 2$ but the derivative of this function is $F ′(x) = - 2sin x+ 2x2 +5x + 1.$

Your answer is correct.

$∫ 2 2 3 5 2 2 sinx + 2x + 5x +1 = 2cosx + 3x + 2 x + x+ c.$
Since $F (0) = - 2 +0 + 0+ 0+ 0 + c = 2 ⇒ c = 4$ so the function
$F (x) = - 2 cosx + 2x3 + 5x2 + x+ 4 3 2$

Question 3

A student wishes to find the area between the curve $8+ 2x- x2$ and the $x$ -axis and produces the following working.

Area	=	$∫ 2 2 - 4(8 +2x - x )dx$
	=	$2 1 3∣∣2 8x+ x - 3 x ∣∣ -4$
	=	$( 8 ) ( - 64 ) 16+ 4- - - - 32 + 16 ----- 3 3$
	=	$72 4 - 3- = - 20$ square units

Which of the following statements are correct.

Your answer is correct.

$8 + 2x - x2 = (4- x)(2+ x)$ so the limits of integration should be -2 and 4. We then get a positive value for the area as expected.

Area	=	$∫ 4 (8 +2x - x2)dx - 2$
	=	$∣∣4 8x+ x2 - 1 x3∣∣ 3 -2$
	=	$( ) ( ) 32+ 16- 64 - - 16 +4 - - 8- 3 3$
	=	$60 - 72= 36 3$ square units

Not correct. Choice (b) is false.

Try again, the anti-derivative is correct.

Not correct. Choice (c) is false.

Try again, the arithmetic is correct.

Not correct. Choice (d) is false.

Try again, the answer is incorrect.

Not correct. Choice (e) is false.

One of the above is the correct answer.

Question 4

We wish to find the area under the curve $y = (x + 1)(x- 1)(x- 3)$ .
Which of the following represents the integrals needed for the calculation?

Not correct. Choice (a) is false.

Try again, this gives $∫ 3 3 x - 3x- x +3 dx -1$ which is not the required integral.

Not correct. Choice (b) is false.

Try again, the limits of integration are incorrect.

Your answer is correct.

The curve looks like this

The value of the integral between -1 and 1 is positive and between 1 and 3 is negative so first we need to integrate between -1 and 1 and then, we need to integrate between 1 and 3 and change that integral’s sign. Thus we evaluate
$∫ 1 3 2 ∫ 3 3 2 -1 x - 3x - x + 3dx- 1 x - 3x - x+ 3 dx$

Not correct. Choice (d) is false.

Try again, you may need to sketch the curve.